nag_dsbevd (f08hcc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dsbevd (f08hcc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dsbevd (f08hcc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the QL or QR algorithm.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dsbevd (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, Integer kd, double ab[], Integer pdab, double w[], double z[], Integer pdz, NagError *fail)

3  Description

nag_dsbevd (f08hcc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band matrix A. In other words, it can compute the spectral factorization of A as
where Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix whose columns are the eigenvectors zi. Thus
Azi=λizi,  i=1,2,,n.

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobNag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_DoNothing or Nag_EigVecs.
3:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of A is stored.
The upper triangular part of A is stored.
The lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     kdIntegerInput
On entry: if uplo=Nag_Upper, the number of superdiagonals, kd, of the matrix A.
If uplo=Nag_Lower, the number of subdiagonals, kd, of the matrix A.
Constraint: kd0.
6:     ab[dim]doubleInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the upper or lower triangle of the n by n symmetric band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[kd+i-j+j-1×pdab], for j=1,,n and i=max1,j-kd,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+kd;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+kd;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[kd+j-i+i-1×pdab], for i=1,,n and j=max1,i-kd,,i.
On exit: ab is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix T are returned in ab using the same storage format as described above.
7:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkd+1.
8:     w[dim]doubleOutput
Note: the dimension, dim, of the array w must be at least max1,n.
On exit: the eigenvalues of the matrix A in ascending order.
9:     z[dim]doubleOutput
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when job=Nag_EigVecs;
  • 1 when job=Nag_DoNothing.
The i,jth element of the matrix Z is stored in
  • z[j-1×pdz+i-1] when order=Nag_ColMajor;
  • z[i-1×pdz+j-1] when order=Nag_RowMajor.
On exit: if job=Nag_EigVecs, z is overwritten by the orthogonal matrix Z which contains the eigenvectors of A. The ith column of Z contains the eigenvector which corresponds to the eigenvalue w[i-1].
If job=Nag_DoNothing, z is not referenced.
10:   pdzIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
  • if job=Nag_EigVecs, pdz max1,n ;
  • if job=Nag_DoNothing, pdz1.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
If fail.errnum=value and job=Nag_DoNothing, the algorithm failed to converge; value elements of an intermediate tridiagonal form did not converge to zero; if fail.errnum=value and job=Nag_EigVecs, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column value/n+1 through value mod n+1.
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_EigVecs, pdz max1,n ;
if job=Nag_DoNothing, pdz1.
On entry, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdz=value.
Constraint: pdz>0.
On entry, pdab=value and kd=value.
Constraint: pdabkd+1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2 = Oε A2 ,
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8  Parallelism and Performance

nag_dsbevd (f08hcc) is not threaded by NAG in any implementation.
nag_dsbevd (f08hcc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The complex analogue of this function is nag_zhbevd (f08hqc).

10  Example

This example computes all the eigenvalues and eigenvectors of the symmetric band matrix A, where
A = 1 2 3 0 0 2 2 3 4 0 3 3 3 4 5 0 4 4 4 5 0 0 5 5 5 .

10.1  Program Text

Program Text (f08hcce.c)

10.2  Program Data

Program Data (f08hcce.d)

10.3  Program Results

Program Results (f08hcce.r)

nag_dsbevd (f08hcc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014